Shock Waves in a One-Dimensional Semi-Infinite Hyperelastic Rod

The excitation of a harmonic wave in a semi–infinite incompressible hyperelastic one-dimensional rod (material according to the Mooney–Rivlin model) leads to the formation and propagation of shock wave fronts arising between half-waves of the initial harmonic wave moving at different speeds. Shock wave fronts lead to the absorption of slow-moving parts by faster ones and, consequently, reduce both kinetic energy and elastic deformation energy with the corresponding release of heat. The explicit Lax–Wendroff difference scheme in combination with the finite element method is used to solve geometrically and physically nonlinear equations of motion.